The Swansea man is being lined up in a £25million bid from the Toffees, who have money to spend after their change of ownership this summer.
Not only is Koeman keen on signing the midfielder, but so too is Steve Walsh, who recently joined the club as director of football after leaving Premier League champions Leicester City.
Sigurdsson was the Swans’ top scorer last season, netting 11 Premier League goals, including one in the 2-1 victory at Goodison Park earlier this year.
The midfielder has regularly been a key player for the Welsh club, whilst also impressing for his nation, helping Iceland get to the quarter-finals of EURO 2016. The 26-year-old scored in the 1-1 draw with Hungary in the group stage.
Koeman kicks off his new season at home to Tottenham Hotspur on August 13th, and is keen to add to his inherited squad in that time. The club are yet to make a marquee signing this summer.